Knowing this first, that there shall come in the last days scoffers, walking after their own lusts... For this they willingly are ignorant of, that by the word of God the heavens were of old,
-2 Peter 3:3-5
[One who scoffs at God's Word.]
By faith Abraham, when he was tried, offered up Isaac: and he that had received the promises offered up his only begotten son, Of whom it was said, That in Isaac shall thy seed be called:
This section says that Isaac was the only begotten son of Abraham, but Abraham had another son (Ishmael) before Isaac.
And Hagar bare Abram a son: and Abram called his son's name, which Hagar bare, Ishmael.
This is an obvious contradiction, therefore the Bible is wrong.
[One who takes God's Word seriously.]
The major significant thing we need to recognize is that the child Ishmael was born to ABRAM
through Hagar, and the child Isaac was born to ABRAHAM
through Sarah. According to Scripture, this is a very big distinction.
And Hagar bare Abram a son: and Abram called his son's name, which Hagar bare, Ishmael. And Abram was fourscore and six years old, when Hagar bare Ishmael to Abram.
Neither shall thy name any more be called Abram, but thy name shall be Abraham; for a father of many nations have I made thee.
For Sarah conceived, and bare Abraham a son in his old age, at the set time of which God had spoken to him.
Without quoting entire chapters, which you can read for yourself, in chapter 16, Sarai was upset that she had not given a son to Abram, and in that culture, it was detrimental for a woman not to bare a child because that was considered their primary purpose. So, in order to give Abram an heir, she offers to give him one through her Egyptian handmaid (Hagar).
In chapter 15, God already promised Abram a son, and God did not need any help with the process, but through a lack of faith, Abram and Sarai choose to bring forth a child through an Egyptian woman. In the New Testament, Paul talks about this situation:
For it is written, that Abraham had two sons, the one by a bondmaid, the other by a freewoman. But he who was of the bondwoman [Ishmael] was born after the flesh; but he of the freewoman [Isaac] was by promise. Which things are an allegory: for these are the two covenants; the one from the mount Sinai, which gendereth to bondage, which is Agar. For this Agar is mount Sinai in Arabia, and answereth to Jerusalem which now is, and is in bondage with her children. But Jerusalem which is above is free, which is the mother of us all. For it is written, Rejoice, thou barren that bearest not; break forth and cry, thou that travailest not: for the desolate hath many more children than she which hath an husband. Now we, brethren, as Isaac was, are the children of promise. But as then he that was born after the flesh persecuted him that was born after the Spirit, even so it is now... So then, brethren, we are not children of the bondwoman, but of the free.
There are many things that can be taken from this section of Scripture (noting that the Bible-hating scoffers will far outnumber the born-again believers, and that the children of the flesh through Ishmael [Muslims] will be at war and hatred against children of promise [Israel]), but most important to this subject is that Isaac is the child of promise through faith. We Christians are born into, and a part of, the children of promise, being born-AGAIN
into a new family; into a different "bloodline" (so to speak) in the spirit, and a different inheritance.
Thus, the children of promise are in a separate lineage, and that lineage was marked with God giving new names unto Abram and Sarai, or in other words, Ishmael was the only begotten of Abram, and Isaac was the only begotten of Abraham. Notice that God (through Paul) also specifically (and very plainly) points out that it is an allegorical concept
allegory (adj): a figurative sentence or discourse; the principal subject is described by another subject resembling it in its properties and circumstances
[(See 'allegory', American Dictionary of the English Language, Noah Webster, 1828, retrieved Oct 2, 2015 [webstersdictionary1828.com])]
Bible-hating scoffers are taking a ridiculous position of attempting to apply literal contradictions to allegorical concepts. That's like taking the book "Animal Farm" by George Orwell, and pointing out that it's a contradiction because animals can't talk. They claim to be so intelligent and educated, but they drop English 101 as soon as they approach the Bible because they hate the True Living God.
Notice that God was the first in the Bible to call Isaac the only son of Abraham:
And he said, Take now thy son, thine only son Isaac, whom thou lovest, and get thee into the land of Moriah; and offer him there for a burnt offering upon one of the mountains which I will tell thee of.
Thus God set up this allegory for us to understand that Isaac is a child of the promise of God, whereas Ishmael is a child of the flesh that is not of God. Isaac and his brethren were meant to follow after God in His righteousness, and Ishmael and his brethren (Muslims) would forever persecute the children of God.
And he [Ishmael] will be a wild man; his hand will be against every man, and every man's hand against him; and he shall dwell in the presence of all his brethren.
As an additional note, the Hebrew language itself is very unique in that is not phonetical like our English language.
phonetic (adj): of or pertaining to speech sounds
[(See 'phonetic', Random House Dictionary, 2014, [www.dictionary.com]; See also Collins English Dictionary, 10th Edition, William Collins Sons & Co, 2009)]
Most languages around the world are based on the sounds or production of a word, and few others, like Chinese, have pictographs (shapes and symbols that depict a meaning). Hebrew is unique in that it is conceptual, meaning that the shape of the individual letters of the alphabet (and their correlating sound) has a conceptual meaning that, when understood, makes one able to read two-thirds of the Hebrew language.
The "Heh" (H) is a breath sound, breathing a burst of air out of your lungs. God conceptually puts the breath (His heart/essence) into them, and represents this in their new names.
Abraham and Sarah are now starting anew in the spirit of God. Whereas Abram and Sarai lacked faith in God, and turned to conceive through an Egyptian, the new Abraham and Sarah repent in faith towards God to conceive a son that will have the promise, and the only begotten son of that union in faith is Isaac.
The Bible clearly says this is an allegory, not literally his only begotten son. It's the only son of the union of Abraham and Sarah in faith. There is no contradiction.